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According to the equation for this completely randomized design, each observation is the sum of three parameters , j , and i( j) . The values of the parameters in the equation are unknown but can be estimated from sample data. The meanings of the terms grand mean, , and treatment effect, j , in the model equation seem fairly clear; the meaning of error effect, i( j) , requires a bit more explanation. Why do observations, Yi j s, in the same treatment level vary from one participant to the next This variation must be due to differences among the participants and to other uncontrolled variables because the parameters and j in the model equation are constants for all participants in the same treatment level. To put it another way, observations in the same treatment

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We refer to finite induced subgraphs of the infinite triangular lattice as hexagon graphs (see Figure 41) A unit disk graph is the intersection graph of disks of equal diameter in the plane They can also be described in terms of distance or proximity models, which consist of a value d 0 and an embedding of nodes in the plane such that (u, v) is an edge if and only if the Euclidean distance between u and v in the specified embedding is at most d For each fixed pair or real values r, s > 0 a graph G can be drawn in R d in an (r, s)civilized manner if its nodes can be mapped to points in R d so that the length of each edge is at most r and the distance between any two points is at least s.

Figure C-10

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An independent (or, stable) set in G is a set V V such that for any u, v V , (u, v) E Note that a proper multicoloring of G is essentially a covering of G with stable sets; each stable set of nodes corresponds to a color and a node v appears in exactly w(v) such sets The weighted chromatic number of a weighted graph (G, w), denoted (G, w), is the smallest number m such that there exists a multicoloring of G of span m, ie, (G, w) is the optimal number of colors required to properly multicolor G Let the weight of a maximal clique in (G, w) be defined as the sum of the weights of the nodes belonging to the clique; note that when G is a hexagon graph, the only maximal cliques are isolated nodes, edges, or triangles.

Computation and Communication Costs. Any protocol being developed for sensor networks should keep in mind the costs associated with computations and communication. With current technology, the cost of computation locally is lower than that of communication in a power-constrained scenario. As a consequence, emphasis should be put on minimizing the communication requirements. Power Constraints. The available power in each sensor is limited by the battery lifetime due to the dif culty or impossibility of recharging the nodes. As a consequence, protocols that tend to minimize the energy consumption or power-aware protocols that adapt to the existing power levels are highly desirable. Additionally, efforts should be made to turn off the nodes themselves, if possible, in the absence of sensing or coordination operations.

The weighted clique number, denoted (G, w) is the maximum weight of any maximal clique in the graph The weighted chromatic number and clique number of a call graph are defined in an analogous fashion For integers k 1, we define the k-locality of a node v to be the induced subgraph consisting of those nodes in G whose graph distance from v is less than or equal to k Given G = (V, E), the graph Gr = (V, E ) is defined by E = E E2 Er 1 Thus any pair of nodes at distance i < r in G is connected by an edge in Gr The problem of channel assignment in a weighted graph (G,w) with reuse distance r is thus the same as multicoloring the graph Gr.

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The unweighted clique number of Gr is the maximum size of any clique in Gr, and is denoted (Gr) Similarly, the chromatic number of Gr, the minimum number of colors needed to color Gr, is denoted by (Gr) It is known that when G is a hexagon graph, (Gr) = (Gr) (see Section 453), and an optimal coloring can be computed in polynomial time We assume that such an optimal coloring of the graph Gr is available; thus, every node in Gr is assigned a color from the set {1, 2, , (Gr)} If G is a hexagon graph, we say that G2 is a square graph Figure 41 shows a base coloring with red, blue, and green (r, b, and g) colors for a hexagon graph It is easy to see that a square graph can be base colored with seven colors; Figure 4.

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